A necessary and sufficient condition of existence of positive solutions for nonlinear singular differential systems

In this paper, by means of monotone iterative technique, a necessary and sufficient condition of the existence of positive solution for a class of nonlinear singular differential system is established, the results of the existence and uniqueness of the positive solution and the iterative sequence of solution are given. In the end, two classes extending boundary value differential systems are discussed and some further results are obtained.     

无穷区间上一类分数阶微分方程正解的显式迭代序列

By applying the monotone iterative method,this study develops two explicit monotone iterative sequences for approximating the minimal and maximal positive solutions.At the same time,by applying the Banach fixed-point theory,an explicit iterative sequence and error estimate for approximating the unique positive solution is obtained.Some examples are given to illustrate the application of the results.     

Successive iterations for unique positive solution of a nonlinear fractional q-integral boundary value problem

In this paper, under certain nonlinear growth conditions, we investigate the existence and successive iterations for the unique positive solution of a nonlinear fractional $q$-integral boundary problem by employing hybrid monotone method, which is a novel approach to nonlinear fractional $q$-difference equation. This paper not only proves the existence of the unique positive solution, but also gives some computable explicit hybrid iterative sequences approximating to the unique positive solution.     

Positive solutions for a class of fractional differential equations with integral boundary conditions

The existence of positive solutions for a class of fractional equations involving the Riemann–Liouville fractional derivative with integral boundary conditions is investigated. By means of the monotone iteration method and some inequalities associated with the Green function, we obtain the existence of a positive solution and establish the iterative sequence for approximating the solution.     

Positive solutions for a nonlinear discrete fractional boundary value problem with a $p$-Laplacian operator

In this paper using the monotone iterative technique we establish the existence and uniqueness of positive solutions for a nonlinear discrete fractional boundary value problem with a $p$-Laplacian operator. Also we discuss an iterative sequence which yields the approximate solution for this problem.     

Banach空间半直线上非线性二阶微分方程组正解的存在性

In this paper, the cone theory and MSnch fixed point theorem combined with the monotone iterative technique are used to investigate the positive solutions for a class of systems of nonlinear singular differential equations with multi-point boundary value conditions on the half line in a Banach space. The conditions for the existence of positive solutions are formulated. In addition, an explicit iterative approximation of the solution is also derived.     

Thompson metric method for solving a class of nonlinear matrix equation

Based on the elegant properties of the Thompson metric, we prove that the general nonlinear matrix equation X^q-A^∗F(X)A=Q(q>1) always has a unique positive definite solution. An iterative method is proposed to compute the unique positive definite solution. We show that the iterative method is more effective as q increases. A perturbation bound for the unique positive definite solution is derived in the end.     

带变号系数的经典Gelfand模型的正解

考察了经典Gelfand模型的正解的存在与迭代,其中非线性项的系数允许在[0,1]中改变符号。利用单调迭代方法得到了一个正解存在定理,给出了相应的迭代程序和收敛速度。由于这个迭代程序是从零函数开始的,因此它是简单、可行并且有效的。     

Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations

By means of a monotone iterative technique, we establish the existence and uniqueness of the positive solutions for a class of higher conjugate-type fractional differential equation with one nonlocal term. In addition, the iterative sequences of solution and error estimation are also given. In particular, this model comes from economics, financial mathematics and other applied sciences, since the initial value of the iterative sequence can begin from an known function, this is simpler and helpful for computation.     

一类带一阶差分的离散问题解的存在性

In this paper, we prove the existence of a positive solution and a negative solution for a class of second order difference equations with dependence on the first order difference. Our proofs are based on the Mountain Pass Lemma and iterative methods.     

矩阵方程X+A*X-nA=I的正定解

In this paper we give some sufficient conditions and some necessary conditions under which the matrix equation X A^*X^-nA=I has a positive definite solution. An iterative method which converges to a positive definite solution of this equation is constructed. And an error estimate formula on this iterative method is also derived.     

2n阶Lidstone边值问题正解的逐次迭代

研究了2n阶Lidstone边值问题正解的逐次迭代,其中非线性项依赖于所有偶数阶导数.通过考察非线性项在某些有国介集合上的“高度”并利用单调迭代方法构造了一个逐次迭代程序.这个迭代程序从一个多项式开始并且是可行的.使用这个结论获得了m个正解的迭代方法,其中m是一个任意的自然数.     

Unique positive solutions for fractional differential equation boundary value problems

In this work, we consider the uniqueness of positive solutions for fractional differential equation boundary value problems. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it.     

Lidstone边值问题正解的单调迭代

考察了Lidstone边值问题正解的单调迭代. 这个迭代程序是从常值函数开始并且是可行的.     

A general iterative scheme for k-strictly pseudo-contractive mappings and optimization problems

In this paper, we introduce a new general iterative scheme for finding fixed points of a strictly pseudo-contractive mapping and then prove that the sequence generated by the proposed iterative scheme converges strongly to a fixed point of the mapping, which is a solution of a certain optimization problem related to a strongly positive bounded linear operator. Additional results of the main result are also obtained.     

Uniqueness and exponential stability of almost periodic positive solution for Lasota‐Wazewska model with impulse and infinite delay

In this paper, impulsive Lasota‐Wazewska model with infinite delay is studied. By using fixed point theorem of decreasing operator, we obtain sufficient conditions for the existence of unique almost periodic positive solution. Particularly, we give iterative sequence, which converges to the almost periodic positive solution. Moreover, we investigate exponential stability of the almost periodic positive solution by Liapunov functional. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence and uniqueness of positive solutions for singular nonlinear elliptic boundary value problems

In this paper we establish the existence and the uniqueness of positive solutions for Dirichlet boundary value problems of nonlinear elliptic equations with singularity. We obtain the existence and the uniqueness by using the mixed monotone method in the cone theory. Moreover, we give an iterative method of constructing the solution. The rate of convergence of the iterative sequence is analyzed.     

一类非线性算子方程的正解问题

研究算子方程X^s+A^*X^-tA=Q的正算子解的存在性问题,通过构造有效的迭代序列,给出了算子方程X^s+A^*X^-tA=Q有正算子解的一些充分条件和必要条件,同时给出了该方程有极大解和唯一解的条件.     

Symmetric positive solutions for second order boundary value problems with integral boundary conditions on time scales

This paper investigates the existence of symmetric positive solutions for a class of nonlinear boundary value problem of second order dynamic equations with integral boundary conditions on time scales. Under suitable conditions, the existence of symmetric positive solution is established by using monotone iterative technique     

Existence and exponential stability of unique almost periodic solution for Lasota–Wazewska red blood cell model with perturbation on time scales

This paper deals with Lasota–Wazewska red blood cell model with perturbation on time scales. By applying the fixed point theorem of decreasing operator, we establish sufficient conditions for the existence of unique almost periodic positive solution. Particularly, we give iterative sequence which converges to the almost periodic positive solution. Moreover, we investigate exponential stability of the almost periodic positive solution by means of Gronwall inequality. Copyright © 2017 John Wiley & Sons, Ltd.